LSA - Latent Semantic Analysis - How to code it in PHP?

LSA links:

• Landauer (co-creator) article on LSA
• the R-project lsa user guide

Here is the complete algorithm. If you have SVD, you are most of the way there. The papers above explain it better than I do.

Assumptions:

• your SVD function will give the singular values and singular vectors in descending order. If not, you have to do more acrobatics.

M: corpus matrix, w (words) by d (documents) (w rows, d columns). These can be raw counts, or tfidf or whatever. Stopwords may or may not be eliminated, and stemming may happen (Landauer says keep stopwords and don't stem, but yes to tfidf).

U,Sigma,V = singular_value_decomposition(M)U:  w x wSigma:  min(w,d) length vector, or w * d matrix with diagonal filled in the first min(w,d) spots with the singular valuesV:  d x d matrixThus U * Sigma * V = M  #  you might have to do some transposes depending on how your SVD code #  returns U and V.  verify this so that you don't go crazy :)

Then the reductionality.... the actual LSA paper suggests a good approximation for the basis is to keep enough vectors such that their singular values are more than 50% of the total of the singular values.

More succintly... (pseudocode)

Let s1 = sum(Sigma).  total = 0for ii in range(len(Sigma)):    val = Sigma[ii]    total += val    if total > .5 * s1:        return ii

This will return the rank of the new basis, which was min(d,w) before, and we'll now approximate with {ii}.

(here, ' -> prime, not transpose)

We create new matrices: U',Sigma', V', with sizes w x ii, ii x ii, and ii x d.

That's the essence of the LSA algorithm.

This resultant matrix U' * Sigma' * V' can be used for 'improved' cosine similarity searching, or you can pick the top 3 words for each document in it, for example. Whether this yeilds more than a simple tf-idf is a matter of some debate.

To me, LSA performs poorly in real world data sets because of polysemy, and data sets with too many topics. It's mathematical / probabilistic basis is unsound (it assumes normal-ish (Gaussian) distributions, which don't makes sense for word counts).

Your mileage will definitely vary.

Tagging using LSA (one method!)

1. Construct the U' Sigma' V' dimensionally reduced matrices using SVD and a reduction heuristic

2. By hand, look over the U' matrix, and come up with terms that describe each "topic". For example, if the the biggest parts of that vector were "Bronx, Yankees, Manhattan," then "New York City" might be a good term for it. Keep these in a associative array, or list. This step should be reasonable since the number of vectors will be finite.

3. Assuming you have a vector (v1) of words for a document, then v1 * t(U') will give the strongest 'topics' for that document. Select the 3 highest, then give their "topics" as computed in the previous step.

This answer isn't directly to the posters' question, but to the meta question of how to autotag news items. The OP mentions Named Entity Recognition, but I believe they mean something more along the line of autotagging. If they really mean NER, then this response is hogwash :)

Given these constraints (600 items / day, 100-200 characters / item) with divergent sources, here are some tagging options:

1. By hand. An analyst could easily do 600 of these per day, probably in a couple of hours. Something like Amazon's Mechanical Turk, or making users do it, might also be feasible. Having some number of "hand-tagged", even if it's only 50 or 100, will be a good basis for comparing whatever the autogenerated methods below get you.

2. Dimentionality reductions, using LSA, Topic-Models (Latent Dirichlet Allocation), and the like.... I've had really poor luck with LSA on real-world data sets and I'm unsatisfied with its statistical basis. LDA I find much better, and has an incredible mailing list that has the best thinking on how to assign topics to texts.

3. Simple heuristics... if you have actual news items, then exploit the structure of the news item. Focus on the first sentence, toss out all the common words (stop words) and select the best 3 nouns from the first two sentences. Or heck, take all the nouns in the first sentence, and see where that gets you. If the texts are all in english, then do part of speech analysis on the whole shebang, and see what that gets you. With structured items, like news reports, LSA and other order independent methods (tf-idf) throws out a lot of information.

Good luck!

(if you like this answer, maybe retag the question to fit it)

That all looks right, up to the last step. The usual notation for SVD is that it returns three matrices A = USV*. S is a diagonal matrix (meaning all zero off the diagonal) that, in this case, basically gives a measure of how much each dimension captures of the original data. The numbers ("singular values") will go down, and you can look for a drop-off for how many dimensions are useful. Otherwise, you'll want to just choose an arbitrary number N for how many dimensions to take.

Here I get a little fuzzy. The coordinates of the terms (words) in the reduced-dimension space is either in U or V, I think depending on whether they are in the rows or columns of the input matrix. Off hand, I think the coordinates for the words will be the rows of U. i.e. the first row of U corresponds to the first row of the input matrix, i.e. the first word. Then you just take the first N columns of that row as the word's coordinate in the reduced space.

HTH

Update:

This process so far doesn't tell you exactly how to pick out tags. I've never heard of anyone using LSI to choose tags (a machine learning algorithm might be more suited to the task, like, say, decision trees). LSI tells you whether two words are similar. That's a long way from assigning tags.

There are two tasks- a) what are the set of tags to use? b) how to choose the best three tags?. I don't have much of a sense of how LSI is going to help you answer (a). You can choose the set of tags by hand. But, if you're using LSI, the tags probably should be words that occur in the documents. Then for (b), you want to pick out the tags that are closest to words found in the document. You could experiment with a few ways of implementing that. Choose the three tags that are closest to any word in the document, where closeness is measured by the cosine similarity (see Wikipedia) between the tag's coordinate (its row in U) and the word's coordinate (its row in U).